Fat identified sounds to me like people who are comfortable being larger. That also seems apparent on the page linked. I'm not a fan of the terminology either but it does make sense.
I don't think it means that nowadays.
Woman identified woman was a phrase that meant women who were no handmaidens of patriarchy, but identified with their own cause.
Nowadays, though, "identifies as woman" means basically everyone who identifies as woman, regardless of reality.
I'd be rather surprised if those safe spaces were designed for actually fat people who are okay with being fat and want to remain so. Actual support groups for fat people would be aware that this would lead to misunderstandings, and therefore wouldn't use it, I think.